So I’ve been travelling a lot in the last month, spending more than half of it outside Portugal. I was in Ottawa, Canada for a Fields Institute workshop, “Categorical Methods in Representation Theory“. Then a little later I was in Erlangen, Germany for one called “Categorical and Representation-Theoretic Methods in Quantum Geometry and CFT“. Despite the similar-sounding titles, these were on fairly different themes, though Marco Mackaay was at both, talking about categorifying the $q$-Schur algebra by diagrams.  I’ll describe the meetings, but for now I’ll start with the first.  Next post will be a summary of the second.

The Ottawa meeting was organized by Alistair Savage, and Alex Hoffnung (like me, a former student of John Baez). Alistair gave a talk here at IST over the summer about a $q$-deformation of Khovanov’s categorification of the Heisenberg Algebra I discussed in an earlier entry. A lot of the discussion at the workshop was based on the Khovanov-Lauda program, which began with categorifying quantum version of the classical Lie groups, and is now making lots of progress in the categorification of algebras, representation theory, and so on.

The point of this program is to describe “categorifications” of particular algebras. This means finding monoidal categories with the property that when you take the Grothendieck ring (the ring of isomorphism classes, with a multiplication given by the monoidal structure), you get back the integral form of some algebra. (And then recover the original by taking the tensor over $\mathbb{Z}$ with $\mathbb{C}$). The key thing is how to represent the algebra by generators and relations. Since free monoidal categories with various sorts of structures can be presented as categories of string diagrams, it shouldn’t be surprising that the categories used tend to have objects that are sequences (i.e. monoidal products) of dots with various sorts of labelling data, and morphisms which are string diagrams that carry those labels on strands (actually, usually they’re linear combinations of such diagrams, so everything is enriched in vector spaces). Then one imposes relations on the “free” data given this way, by saying that the diagrams are considered the same morphism if they agree up to some local moves. The whole problem then is to find the right generators (labelling data) and relations (local moves). The result will be a categorification of a given presentation of the algebra you want.

So for instance, I was interested in Sabin Cautis and Anthony Licata‘s talks connected with this paper, “Heisenberg Categorification And Hilbert Schemes”. This is connected with a generalization of Khovanov’s categorification linked above, to include a variety of other algebras which are given a similar name. The point is that there’s such a “Heisenberg algebra” associated to different subgroups $\Gamma \subset SL(2,\mathbf{k})$, which in turn are classified by Dynkin diagrams. The vertices of these Dynkin diagrams correspond to some generators of the Heisenberg algebra, and one can modify Khovanov’s categorification by having strands in the diagram calculus be labelled by these vertices. Rules for local moves involving strands with different labels will be governed by the edges of the Dynkin diagram. Their paper goes on to describe how to represent these categorifications on certain categories of Hilbert schemes.

Along the same lines, Aaron Lauda gave a talk on the categorification of the NilHecke algebra. This is defined as a subalgebra of endomorphisms of $P_a = \mathbb{Z}[x_1,\dots,x_a]$, generated by multiplications (by the $x_i$) and the divided difference operators $\partial_i$. There are different from the usual derivative operators: in place of the differences between values of a single variable, they measure how a function behaves under the operation $s_i$ which switches variables $x_i$ and $x_{i+1}$ (that is, the reflection in the hyperplane where $x_i = x_{i+1}$). The point is that just like differentiation, this operator – together with multiplication – generates an algebra in $End(\mathbb{Z}[x_1,\dots,x_a]$. Aaron described how to categorify this presentation of the NilHecke algebra with a string-diagram calculus.

So anyway, there were a number of talks about the explosion of work within this general program – for instance, Marco Mackaay’s which I mentioned, as well as that of Pedro Vaz about the same project. One aspect of this program is that the relatively free “string diagram categories” are sometimes replaced with categories where the objects are bimodules and morphisms are bimodule homomorphisms. Making the relationship precise is then a matter of proving these satisfy exactly the relations on a “free” category which one wants, but sometimes they’re a good setting to prove one has a nice categorification. Thus, Ben Elias and Geordie Williamson gave two parts of one talk about “Soergel Bimodules and Kazhdan-Lusztig Theory” (see a blog post by Ben Webster which gives a brief intro to this notion, including pointing out that Soergel bimodules give a categorification of the Hecke algebra).

One of the reasons for doing this sort of thing is that one gets invariants for manifolds from algebras – in particular, things like the Jones polynomial, which is related to the Temperley-Lieb algebra. A categorification of it is Khovanov homology (which gives, for a manifold, a complex, with the property that the graded Euler characteristic of the complex is the Jones polynomial). The point here is that categorifying the algebra lets you raise the dimension of the kind of manifold your invariants are defined on.

So, for instance, Scott Morrison described “Invariants of 4-Manifolds from Khonanov Homology“.  This was based on a generalization of the relationship between TQFT’s and planar algebras.  The point is, planar algebras are described by the composition of diagrams of the following form: a big circle, containing some number of small circles.  The boundaries of each circle are labelled by some number of marked points, and the space between carries curves which connect these marked points in some way.  One composes these diagrams by gluing big circles into smaller circles (there’s some further discussion here including a picture, and much more in this book here).  Scott Morrison described these diagrams as “spaghetti and meatball” diagrams.  Planar algebras show up by associating a vector spaces to “the” circle with $n$ marked points, and linear maps to each way (up to isotopy) of filling in edges between such circles.  One can think of the circles and marked-disks as a marked-cobordism category, and so a functorial way of making these assignments is something like a TQFT.  It also gives lots of vector spaces and lots of linear maps that fit together in a particular way described by this category of marked cobordisms, which is what a “planar algebra” actually consists of.  Clearly, these planar algebras can be used to get some manifold invariants – namely the “TQFT” that corresponds to them.

Scott Morrison’s talk described how to get invariants of 4-dimensional manifolds in a similar way by boosting (almost) everything in this story by 2 dimensions.  You start with a 4-ball, whose boundary is a 3-sphere, and excise some number of 4-balls (with 3-sphere boundaries) from the interior.  Then let these 3D boundaries be “marked” with 1-D embedded links (think “knots” if you like).  These 3-spheres with embedded links are the objects in a category.  The morphisms are 4-balls which connect them, containing 2D knotted surfaces which happen to intersect the boundaries exactly at their embedded links.  By analogy with the image of “spaghetti and meatballs”, where the spaghetti is a collection of 1D marked curves, Morrison calls these 4-manifolds with embedded 2D surfaces “lasagna diagrams” (which generalizes to the less evocative case of “$(n,k)$ pasta diagrams”, where we’ve just mentioned the $(2,1)$ and $(4,2)$ cases, with $k$-dimensional “pasta” embedded in $n$-dimensional balls).  Then the point is that one can compose these pasta diagrams by gluing the 4-balls along these marked boundaries.  One then gets manifold invariants from these sorts of diagrams by using Khovanov homology, which assigns to

Ben Webster talked about categorification of Lie algebra representations, in a talk called “Categorification, Lie Algebras and Topology“. This is also part of categorifying manifold invariants, since the Reshitikhin-Turaev Invariants are based on some monoidal category, which in this case is the category of representations of some algebra.  Categorifying this to a 2-category gives higher-dimensional equivalents of the RT invariants.  The idea (which you can check out in those slides) is that this comes down to describing the analog of the “highest-weight” representations for some Lie algebra you’ve already categorified.

The Lie theory point here, you might remember, is that representations of Lie algebras $\mathfrak{g}$ can be analyzed by decomposing them into “weight spaces” $V_{\lambda}$, associated to weights $\lambda : \mathfrak{g} \rightarrow \mathbf{k}$ (where $\mathbf{k}$ is the base field, which we can generally assume is $\mathbb{C}$).  Weights turn Lie algebra elements into scalars, then.  So weight spaces generalize eigenspaces, in that acting by any element $g \in \mathfrak{g}$ on a “weight vector” $v \in V_{\lambda}$ amounts to multiplying by $\lambda{g}$.  (So that $v$ is an eigenvector for each $g$, but the eigenvalue depends on $g$, and is given by the weight.)  A weight can be the “highest” with respect to a natural order that can be put on weights ($\lambda \geq \mu$ if the difference is a nonnegative combination of simple weights).  Then a “highest weight representation” is one which is generated under the action of $\mathfrak{g}$ by a single weight vector $v$, the “highest weight vector”.

The point of the categorification is to describe the representation in the same terms.  First, we introduce a special strand (which Ben Webster draws as a red strand) which represents the highest weight vector.  Then we say that the category that stands in for the highest weight representation is just what we get by starting with this red strand, and putting all the various string diagrams of the categorification of $\mathfrak{g}$ next to it.  One can then go on to talk about tensor products of these representations, where objects are found by amalgamating several such diagrams (with several red strands) together.  And so on.  These categorified representations are then supposed to be usable to give higher-dimensional manifold invariants.

Now, the flip side of higher categories that reproduce ordinary representation theory would be the representation theory of higher categories in their natural habitat, so to speak. Presumably there should be a fairly uniform picture where categorifications of normal representation theory will be special cases of this. Vlodymyr Mazorchuk gave an interesting talk called 2-representations of finitary 2-categories.  He gave an example of one of the 2-categories that shows up a lot in these Khovanov-Lauda categorifications, the 2-category of Soergel Bimodules mentioned above.  This has one object, which we can think of as a category of modules over the algebra $\mathbb{C}[x_1, \dots, x_n]/I$ (where I  is some ideal of homogeneous symmetric polynomials).  The morphisms are endofunctors of this category, which all amount to tensoring with certain bimodules – the irreducible ones being the Soergel bimodules.  The point of the talk was to explain the representations of 2-categories $\mathcal{C}$ – that is, 2-functors from $\mathcal{C}$ into some “classical” 2-category.  Examples would be 2-categories like “2-vector spaces”, or variants on it.  The examples he gave: (1) [small fully additive $\mathbf{k}$-linear categories], (2) the full subcategory of it with finitely many indecomposible elements, (3) [categories equivalent to module categories of finite dimensional associative $\mathbf{k}$-algebras].  All of these have some claim to be a 2-categorical analog of [vector spaces].  In general, Mazorchuk allowed representations of “FIAT” categories: Finitary (Two-)categories with Involutions and Adjunctions.

Part of the process involved getting a “multisemigroup” from such categories: a set $S$ with an operation which takes pairs of elements, and returns a subset of $S$, satisfying some natural associativity condition.  (Semigroups are the case where the subset contains just one element – groups are the case where furthermore the operation is invertible).  The idea is that FIAT categories have some set of generators – indecomposable 1-morphisms – and that the multisemigroup describes which indecomposables show up in a composite.  (If we think of the 2-category as a monoidal category, this is like talking about a decomposition of a tensor product of objects).  So, for instance, for the 2-category that comes from the monoidal category of $\mathfrak{sl}(2)$ modules, we get the semigroup of nonnegative integers.  For the Soergel bimodule 2-category, we get the symmetric group.  This sort of thing helps characterize when two objects are equivalent, and in turn helps describe 2-representations up to some equivalence.  (You can find much more detail behind the link above.)

On the more classical representation-theoretic side of things, Joel Kamnitzer gave a talk called “Spiders and Buildings”, which was concerned with some geometric and combinatorial constructions in representation theory.  These involved certain trivalent planar graphs, called “webs”, whose edges carry labels between 1 and $(n-1)$.  They’re embedded in a disk, and the outgoing edges, with labels $(k_1, \dots, k_m)$ determine a representation space for a group $G$, say $G = SL_n$, namely the tensor product of a bunch of wedge products, $\otimes_j \wedge^{k_j} \mathbb{C}^n$, where $SL_n$ acts on $\mathbb{C}^n$ as usual.  Then a web determines an invariant vector in this space.  This comes about by having invariant vectors for each vertex (the basic case where $m =3$), and tensoring them together.  But the point is to interpret this construction geometrically.  This was a bit outside my grasp, since it involves the Langlands program and the geometric Satake correspondence, neither of which I know much of anything about, but which give geometric/topological ways of constructing representation categories.  One thing I did pick up is that it uses the “Langlands dual group” $\check{G}$ of $G$ to get a certain metric space called $Gn_{\check{G}}$.  Then there’s a correspondence between the category of representations of $G$ and the category of (perverse, constructible) sheaves on this space.  This correspondence can be used to describe the vectors that come out of these webs.

Jim Dolan gave a couple of talks while I was there, which actually fit together as two parts of a bigger picture – one was during the workshop itself, and one at the logic seminar on the following Monday. It helped a lot to see both in order to appreciate the overall point, so I’ll mix them a bit indiscriminately. The first was called “Dimensional Analysis is Algebraic Geometry”, and the second “Toposes of Quasicoherent Sheaves on Toric Varieties”. For the purposes of the logic seminar, he gave the slogan of the second talk as “Algebraic Geometry is a branch of Categorical Logic”. Jim’s basic idea was inspired by Bill Lawvere’s concept of a “theory”, which is supposed to extend both “algebraic theories” (such as the “theory of groups”) and theories in the sense of physics.  Any given theory is some structured category, and “models” of the theory are functors into some other category to represent it – it thus has a functor category called its “moduli stack of models”.  A physical theory (essentially, models which depict some contents of the universe) has some parameters.  The “theory of elastic scattering”, for instance, has the masses, and initial and final momenta, of two objects which collide and “scatter” off each other.  The moduli space for this theory amounts to assignments of values to these parameters, which must satisfy some algebraic equations – conservation of energy and momentum (for example, $\sum_i m_i v_i^{in} = \sum_i m_i v_i^{out}$, where $i \in 1, 2$).  So the moduli space is some projective algebraic variety.  Jim explained how “dimensional analysis” in physics is the study of line bundles over such varieties (“dimensions” are just such line bundles, since a “dimension” is a 1-dimensional sort of thing, and “quantities” in those dimensions are sections of the line bundles).  Then there’s a category of such bundles, which are organized into a special sort of symmetric monoidal category – in fact, it’s contrained so much it’s just a graded commutative algebra.

In his second talk, he generalized this to talk about categories of sheaves on some varieties – and, since he was talking in the categorical logic seminar, he proposed a point of view for looking at algebraic geometry in the context of logic.  This view could be summarized as: Every (generalized) space studied by algebraic geometry “is” the moduli space of models for some theory in some doctrine.  The term “doctrine” is Bill Lawvere’s, and specifies what kind of structured category the theory and the target of its models are supposed to be (and of course what kind of functors are allowed as models).  Thus, for instance, toposes (as generalized spaces) are supposed to be thought of as “geometric theories”.  He explained that his “dimensional analysis doctrine” is a special case of this.  As usual when talking to Jim, I came away with the sense that there’s a very large program of ideas lurking behind everything he said, of which only the tip of the iceberg actually made it into the talks.

Next post, when I have time, will talk about the meeting at Erlangen…